A) \[\frac{\pi }{3}\]
B) \[\frac{2\pi }{3}\]
C) \[\frac{\pi }{6}\]
D) \[\frac{\pi }{4}\]
Correct Answer: B
Solution :
The given progressive waves are \[{{y}_{1}}=a\sin \,(\omega t+{{\phi }_{1}})\] \[{{y}_{2}}=a\sin \,(\omega t+{{\phi }_{2}})\] The resultant of two waves is \[y={{y}_{1}}+{{y}_{2}}\] \[=a\,[\sin \,(\omega t+{{\phi }_{1}})+\sin \,(\omega t+{{\phi }_{2}})]\] If A is the amplitude of resultant wave, then A = a (given) \[\therefore {{A}^{2}}={{a}^{2}}+{{a}^{2}}+2{{a}^{2}}\cos \phi \] \[or{{a}^{2}}={{a}^{2}}+{{a}^{2}}+2{{a}^{2}}\cos \phi \] \[or\cos \phi =-\frac{1}{2}=\cos {{120}^{o}}\] \[\therefore \phi ={{120}^{o}}=\frac{2\pi }{3}\] \[Thus,\,\,{{\phi }_{1}}-{{\phi }_{2}}=\frac{2\pi }{3}\]
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